1. Define a sequence. A sequence is an ordered list of numbers following a particular pattern. Example: 2, 4, 6, 8, ... 2. What is an arithmetic sequence? An arithmetic sequence is a sequence where the difference between consecutive terms is constant. Example: 3, 5, 7, 9, ... (common difference 2) 3. What is the common difference of an arithmetic sequence? The common difference $d$ is the constant amount added to each term to get the next term. 4. 2, 4, 6, 8, ... are in A.P. Find 9th term. Formula: $a_n = a_1 + (n-1)d$ Here, $a_1=2$, $d=2$, $n=9$ $$a_9 = 2 + (9-1)\times 2 = 2 + 16 = 18$$ 5. Which term of A.P. 5, 2, -1, ... is -85? $a_1=5$, $d=2-5=-3$, find $n$ such that $a_n=-85$ $$-85 = 5 + (n-1)(-3)$$ $$-85 - 5 = -3(n-1)$$ $$-90 = -3(n-1)$$ $$n-1 = 30 \Rightarrow n=31$$ 6. Find the number of terms in the arithmetic progression if $a_1=3$, $d=7$ and $a_n=59$. $$59 = 3 + (n-1)7$$ $$59 - 3 = 7(n-1)$$ $$56 = 7(n-1)$$ $$n-1 = 8 \Rightarrow n=9$$ 7. If $a_{n-3} = 2n - 5$, find the nth term of the sequence. Replace $n$ by $n-3$: $$a_{n-3} = 2n - 5$$ Let $k = n-3$, then $a_k = 2(k+3) - 5 = 2k + 6 - 5 = 2k + 1$ So, $$a_n = 2n + 1$$ 8. If $a_{n-2} = 3n - 11$, find $a_n$. Let $k = n-2$, then $$a_k = 3(k+2) - 11 = 3k + 6 - 11 = 3k - 5$$ So, $$a_n = 3n - 5$$ 9. Find the first two terms of the sequence whose rth term is $3r + 1$. $$a_1 = 3(1) + 1 = 4$$ $$a_2 = 3(2) + 1 = 7$$ 10. Find the first three terms of sequence if $a_n - a_{n-1} = n + 1$ and $a_4 = 14$. We have $a_n = a_{n-1} + n + 1$ Calculate backwards: $$a_4 = 14$$ $$a_4 - a_3 = 4 + 1 = 5 \Rightarrow a_3 = 14 - 5 = 9$$ $$a_3 - a_2 = 3 + 1 = 4 \Rightarrow a_2 = 9 - 4 = 5$$ $$a_2 - a_1 = 2 + 1 = 3 \Rightarrow a_1 = 5 - 3 = 2$$ First three terms: 2, 5, 9 11. Find the 13th term of the sequence $x, 12 - x, ...$ Assuming arithmetic progression: $$a_1 = x, a_2 = 12 - x$$ Common difference: $$d = (12 - x) - x = 12 - 2x$$ $$a_{13} = a_1 + 12d = x + 12(12 - 2x) = x + 144 - 24x = 144 - 23x$$ 12. Find the 5th term of the G.P. 3, 6, 12, ... Formula: $a_n = a_1 r^{n-1}$ Here, $a_1=3$, $r=\frac{6}{3}=2$ $$a_5 = 3 \times 2^{4} = 3 \times 16 = 48$$ 13. Find the sum of the infinite geometric sequence $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$ Sum formula for infinite G.P. with $|r|<1$: $$S = \frac{a}{1-r}$$ Here, $a=\frac{1}{2}$, $r=\frac{1}{2}$ $$S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1$$ 14. Find the sum of infinite geometric series $2 + 1 + 0.5 + ...$ Here, $a=2$, $r=\frac{1}{2}$ $$S = \frac{2}{1 - \frac{1}{2}} = \frac{2}{\frac{1}{2}} = 4$$ 15. Convert the recurring decimal 0.23 (where 23 repeats) into an equivalent common fraction. Let $x = 0.232323...$ Multiply by 100: $$100x = 23.2323...$$ Subtract: $$100x - x = 23.2323... - 0.2323... = 23$$ $$99x = 23 \Rightarrow x = \frac{23}{99}$$ 16. Write first five terms of the sequence if $a_n = n((-1)^2 + n^2)$. Since $(-1)^2 = 1$, $$a_n = n(1 + n^2) = n + n^3$$ Calculate for $n=1$ to 5: $$a_1 = 1 + 1 = 2$$ $$a_2 = 2 + 8 = 10$$ $$a_3 = 3 + 27 = 30$$ $$a_4 = 4 + 64 = 68$$ $$a_5 = 5 + 125 = 130$$ 17. List the first 10 terms of the sequence that begins with 2 and in which each successive term is 3 more than the preceding term. This is an A.P. with $a_1=2$, $d=3$ $$a_n = 2 + (n-1)3 = 3n - 1$$ Terms: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29 18. Find the value of $\sum_{j=0}^5 (2^{j+1} - 2!)$. Note $2! = 2$ $$\sum_{j=0}^5 (2^{j+1} - 2) = \sum_{j=0}^5 2^{j+1} - \sum_{j=0}^5 2 = \sum_{k=1}^6 2^k - 6 \times 2$$ Sum of powers: $$\sum_{k=1}^6 2^k = 2(2^6 - 1) = 2(64 - 1) = 2 \times 63 = 126$$ Subtract: $$126 - 12 = 114$$ 19. If $S = \{1, 3, 5\}$, find the value of $\sum_{j \in S} \frac{1}{j}$. $$\frac{1}{1} + \frac{1}{3} + \frac{1}{5} = 1 + \frac{1}{3} + \frac{1}{5} = \frac{15}{15} + \frac{5}{15} + \frac{3}{15} = \frac{23}{15}$$ 20. Compute the double sum $\sum_{j=1}^3 \sum_{i=0}^4 j^3 i^2$. Calculate inner sum: $$\sum_{i=0}^4 i^2 = 0 + 1 + 4 + 9 + 16 = 30$$ Calculate outer sum: $$\sum_{j=1}^3 j^3 = 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36$$ Double sum: $$36 \times 30 = 1080$$